Cut-Set

By removing certain branches of a graph, a connected graph can be separated into two parts. This operation is equivalent to cutting a graph into two parts. The term cut-set is derived from this property.

Consider the graph shown in Fig. a. Let us remove branches 1, 3, 4 and 5 from this graph. After removal of these branches we see that the connected graph of Fig. a is separated into two distinct parts each of which is connected as shown in Fig. b. Now suppose each removed branch is replaced one at a time. Fig.c shows the result if branch 1 is replaced in Fig. b. The graph is now connected. Similarly, if we replace the removed branches 3, 4 and 5 of the set {1, 3,4, 5} one at a time, all other ones remaining removed, we get the resulting graphs shown in Figs. d, e and f. The set formed by the branches 1, 3,

FIG.

4 and 5 is called the cut-set of the connected graph of Fig. a. It is represented by {1, 3, 4, 5} where the numerals 1, 3, 4, 5 enclosed by the curl bracket, stand for the respective branches of the cut-set. We can define a cut-set as follows:

A cut-set is a set of minimum number of branches of a connected graph whose retrieval causes the graph to be cut (separated) into exactly two parts (sub graphs), each of which is a connected^ graph, with the condition that replacing any one branch from the cut-set renders the graph connectedly.

This implies that if any branch of the cut-set is not removed (deleted), the graph remains connected. In other words, restoring any one of the branches of the cut-set will destroy the separation property of the two parts.

For testing whether a set of branches of a connected graph constitute a cut-set we have replaced each removed branch one at a time, while we have kept all the other branches of the set removed. Then we have seen that the resulting graph is connected or not. This procedure is very lengthy for testing a set of branches to determine if it is a cut-set. A more convenient method is to check the condition that removal of any of the proper subset of this set (called cut-set) leaves the graph connected. Let us consider the set {1, 3, 4, 5}. Some of its subsets are (1, 3, 4), (3, 4, 5), (1, 4, 5). If the subset (1, 3,4) is removed from the graph of Fig.a, the resulting graph is shown in Fig. f. Thus, removal of the subset (1, 3, 4) does not cut the graph into two parts. Similarly, if the subsets (3, 4, 5) and (1, 4, 5) are removed from the graph, the resulting graphs are shown in Fig.c and d respectively. In both these cases also the graphs are not cut into two distinct parts. Thus, we can also define a cut-set as follows:

A cut-set is a set of minimum number of branches of a connected graph whose removal causes the graph to be cut into exactly two connected sub graphs with the restriction that the removal of any of the proper subset of this set (called cut-set) reunites the graph into a single connected graph. Each cut-set contains one and only one twig, the remaining elements being tree links.

A cut-set is shown on a graph by a dashed line, where the dashed line passes through the brandies defining the cut-set. A graph can have more than one cut-set. Also, every graph has at least one cut-set.

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